GEOM Ivy, and that by any parallel plane, is an ellipse ; and the surface may be considered as generated by a variable ellipse moving parallel to itself along the two hypcrbolas as direct-riees. In the hypcrboloid of two sheets (fig. 25), the sec- tions by the planes of :x and ::y are the liyperbolas SOLID .-‘?T.LY'1'ICAL.] ea in z3 9:2 z2 '1/2 -7.‘ —..'=1 7 '7» "‘T=1: C‘ a" c‘ I)" having the common tians- verse axis :02’; the section by any plane :4 = :t 7 par- allel to that of my, 7 being in absolute magnitude > c, is the ellipse T 1/ . . Fi .24. and the surface, consisting g of tvo distinct portions or sheets, may be considered as generated by a variable ellipse moving __ parallel to itself along the “ liyperbolas as direetriees. 7 The hyperbolic paraboloid is such (and it is easy from the figure to understand how this may be the case) that there exist upon it two singly infinite seiies of right lines. The same is the case with the hyperboloid of one sheet (ruled or skew hyperboloid, as with reference to this property it is termed). If we imagine two equal and parallel circular disks, their points connected by strings of equal length, so that these are the generatiiig lines of a right eircular cylinder, then by turn- ing one of the disks about its centre through the same angle in one or the other direction, " the strings will in each case generate one and the same liyperboloid, and will in regard to it be the two systems of lines on the surface, or say the two systems of generat- ing lines ; and the general configuration is the same when instead of circles we have ellipses. It has been already shown analytically 0 Fig. 25. that the cquationfiz + — = 1 is satisfied by each of two pairs of linear relations between the coordinates. C-'zu'L'cs; Tuizgcizf, Osculating Plane, C’-at-7'vaI‘m'e, (ST. 38. It will be convenient to consider the coordinates (.r, 3/, 2) of the point on the curve as given in terms of a parameter 0, so that d.c, dy, dz, (l'~’.L', &c., will be proper- de’ do’ do’ (Z92, °' analytical formulae will be given. rent coordinates. The tangent is the line through the point (av, 3/, 2) and the consecutive point + dx, -3} + dy, z + dz) ; its equations therefore are tional to But only a part of the .5, 77, § are used as ciir- E -96 =n—y= §—_z _ (1.2: (I3; (I: The osculating plane is the pla.iie through the point and two consecutive points, and contains therefore the tangent; its equation is £‘xs 71”‘?/7 (-3 (Lo , dy dz dgac , cl“‘y , (l“’z or, what is the same thing, t’-E-=L’)(d.7jd9-”--(326323/) + (-q—y)(dzd’:z:—d:cd9z) + (g‘— :)(d:z:d5'y—tIyd’x) = O. _ The normal plane is the plane through the point at right angles to the tangent. It meets the osculating plane in a line called the principal normal ; and drawing through the point a line at right angles to the osculating plane, this is called the binorinal. Ve have thus at the point a =0 ETRY 419 set of three rectangular axes—the tangent, the principal normal, and the binomial. “'0 have through the point and three consecutive points a sphere of spherical curvature,—the centre and radius thereof being the centre, and radius, of spherical curvature. The sphere is met by the osculating plane in the circle of absolute curvature,—tlie centre a.iid radius thereof being the centre, and radius, of absolute curvature. The centre of absolute curvature is also the intersection of the principal normal by the normal plane at the consecutive point. Surfaces; Tangent Lines and Plane, Cwvaturc, (ye. 39. It will be convenient to consider the surface as given by an equation f(.7c, 3/, between the coordi- nates ; taking (as, y, 2) for the coordinates of a given point, and (x+d.v, 3/+d_I/, z+dz) for those of a consecutive point, the increments (Ix, (Iy, dz satisfy the condition fig: % dy + dz = 0 , but the ratio of two of the increments, suppose dz: dy, may be regarded as arbitrary. Only a part of the analy- tical formulac will be given. 5, 77, § are used as current coordinates. Ve have through the point a singly infinite series oi right lines, each meeting the surface in a consecutive point, or say having each of them two-point intersection with the surface. These lines lie all of them in a plane which is the tangent plane ; its equation is
- £,<s—w> + j§(n—y)+j§§((—=) = 0,
as is a.t once verified by observing that this equation is satisfied (irrespectively of the value of the : dy) on writing therein 5, 77, §=x + dx, 3/ + 113/, z + :72. The line through the point at right angles to the tan- gent plane is called the normal ; its equations are £1” ,, Q = (1 . clf df df In the series of tangent lines there are in general two (real or imaginary) lines, each of which meets the surface in a second consecutive point, or say it has three—point intersection with the surface; these are called the chief- tangents (Haiipt—taiigeiiten). The tangent-plane cuts the surface in a curve, having at the point of contact a node (double point), the tangents to the two branches being the chief-tangents. In the case of a quadrie surface the curve of intersec- tion, qua curve of the second order, can only have a node by breaking 11 into a pair of lines; that is, every tangent- plane meets the surface in a pair of lines, or we lia.ve on the surface two singly infinite systems of lines ,' these are real for the hyperbolic paraboloid and the hyperboloid of one sheet, imaginary in other cases. At each point of a surface the chief-tangents determine two directions; and passing along one of them to a con- secutive point, and thence (Without abrupt change of direction) along the new chief-taiigent to a consecutive point, and so on, we have on the surface a cliief-tangent curve ; and there a.re, it is clear, two singly infinite series of such curves. In the case of a quadric surface, the curves a.re the right lines on the surface. 40. If at the point we draw in the tangent—plane two lines bisectiiig the angles between the chief-tangents, these lines (which are a.t right angles to each other) are called the principal tangents.‘ Ve have thus at each point of (la: + 1 The point on the surface may be such that the directions of the principal tangents become arbitrary; the point is then an iimbilicus. It is in the text assumed that the point on the surface is not an
unibilicus.